THE LEAP MINUTE-PREDICTING THE UNPREDICTABLE - Hanksville
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(Preprint) AAS 13-510
THE LEAP MINUTE—PREDICTING THE UNPREDICTABLE
John H. Seago*
Amidst the controversial effort to eliminate leap seconds from Coordinated Uni-
versal Time (UTC), the so-called leap minute has been recurrently tendered as
an alternative approach to reconcile atomic time with astronomical time-of-day.
This paper discourses some civil-timekeeping requirements addressed by inter-
calary minutes, factors that could affect leap-minute scheduling, and the sup-
posed advantages and disadvantages of such a compromise proposal. Particular-
ly, the leap minute does not appear to have obvious and overwhelming ad-
vantages over the convention it proposes to replace; furthermore, the inaugural
leap minute is expected to happen beyond the professional lifetimes of current
advocates, and there is no evidence that its official adoption now would ensure
its operational acceptance later.
INTRODUCTION
Within the context of a contentious debate to redefine Coordinated Universal Time (UTC), a
wide range of spectators have suggested the so-called leap minute as an alternative method for
reconciling atomic time with Earth rotation. Apparent supporters include a former director of the
International Bureau of Weights and Measures (BIPM), specialists in the timekeeping industry,
unnamed officials associated with the Radiocommunication Sector of International Telecommu-
nication Union (ITU-R), expert consumers of civil time, interested journalists, technology blog-
gers, and some members of the general public. Popular press reports have further echoed the
promotion of leap minutes by those closely associated with the process that led to a call for more
studies by the 2012 ITU-R Radiocommunication Assembly.1, 2 A subsequent Bloomberg editorial
expressed the situation this way:†
“Several years ago, some scientists suggested scheduling a leap hour for the year 2600.
This idea was abandoned as impractical, given that the instructions would have to be left
for people six centuries hence. But could there instead be, say, a leap minute every half
century?”
PERSPECTIVES REGARDING THE LEAP MINUTE
Observant Citizenry
The leap minute—a clock adjustment introduced when its difference from UT1 approach one-
half minute— is often perceived as a practical compromise by followers of the debate over UTC
redefinition. Examples of supportive commentary (with added highlighting) go like this:
*
Astrodynamics Engineer, Analytical Graphics, Inc., 220 Valley Creek Blvd, Exton, PA, 19341-2380.
†
http://www.bloomberg.com/news/2012-02-29/time-is-of-the-essence-in-deciding-fate-of-the-leap-second-view.html
1
• “…a 30-second discrepancy between the clocks and the astronomical noon wouldn’t hurt any-
one.”*
• “The average person would not notice if sunrise is off by 30 seconds.”†
• “Maybe we should wait 100 years and then have a leap minute.”‡
• “…I wonder why we don’t use leap minutes rather than leap seconds. The Earth’s elliptical orbit
already causes the Sun to appear to move up to 15 minutes slower or faster than mean solar time.
An additional variance of one minute from true mean solar time would not be a problem for the av-
erage person, and correcting clocks by one minute once or a couple of times a century would be
much easier for the engineers to keep track of than these continual one-second corrections. And a
leap minute would be much bigger news than a leap second.Ӥ
• “Alternatively, rather than abandon leap-seconds make it leap-minutes. Once in a century we
could probably use an extra minute anyway.”**
Specialist Consumers
In response to a 2011 questionnaire conducted by the IERS Earth Orientation Center, the no-
tion of leap minutes was mentioned a dozen times by those both favoring and opposing that no-
tion.3 Comments (with added highlighting) included:
• “Leap minutes or leap hours would be very disruptive.”
• “Perhaps, a ‘leap minute’ once a century might do. That would be better than this silly idea of a
‘leap hour’.”
• “Why not introducing leap minutes instead of leap seconds?” [as an alternative proposal]
• “I am wondering there has been enough discussion regarding introducing ‘leap minute’ instead of
leap second.” [as an alternative proposal]
• “But if we want follow day and night variation, then within decades we’ll need a leap minute or
within millennia a leap hour... Are these any better than the leap seconds?”
• “Alternatively, the concept would remain for DUT1 but change only when added up to a ‘leap mi-
nute’.”
• “A more realistic option [than a leap hour] with less undesirable effects would be a ‘leap minute’,
but that would also defer difficult issues irresponsibly.”
• “…millennia into the future, it might be more logical to insert a leap minute, or better yet, perhaps
once a century make accurate clocks that run just a bit slower, thus redefining the length of the sec-
ond.”
• “the small and predictable leap second increments are much more tolerable than larger step adjust-
ments proposed (leap minute or leap hour) and less troubling…”
• “I prefer ‘leap minute’ introduced every 50 or 100 years.” [as an alternative proposal]
*
http://motls.blogspot.com/2012/01/leap-seconds-may-be-abolished-in-2015.html
†
http://www.realfreemarket.org/blog/2012/01/22/leap-second-should-be-leap-minute/
‡
http://phys.org/news/2012-06-added.html
§
Barreiro, A., (2012), “Why not leap minutes?” Comment on the article “A Glitch in Time.”
http://www.skyandtelescope.com/community/skyblog/newsblog/A-Glitch-in-Time-160824935.html
**
http://mm.icann.org/pipermail/tz/1999-December/010734.html
2
• “…the issue is a problem that should not be left for future generations (leap minutes, for exam-
ple).”
• “A leap minute could be introduced…” [as an alternative proposal]
Of 21 responses that favored another preference besides either the status quo or “UTC without
leap seconds”, four (4) specifically proposed leap minutes as an alternative proposal.
Experts and Officials
The insertion of a leap minute into UTC “in about fifty years” was notably advanced by ex-
perts in 2001 on the supposition that it would be “relatively easy to adopt.”4 However, even be-
fore the first leap second in 1972, the leap minute had already been contemplated as a potential
adjustment mechanism for civil time based on atomic frequency:5
Everyday users would not need to be concerned about the introduction of an occasionally
modified, atomic scale of time. Various local universal times, standardized by conven-
tion, such as Mountain Standard Time (MST), differ from GMT by an integral number of
hours, depending on locations. Such local time scales are often adjusted periodically by
one-hour steps to yield “daylight savings time,” or to return to “standard” time. Similar
local times could be derived just as well from an atomic scale, by similar adjustments and
zoning, and by the deletion, every few years, of a small number of “one-second leaps” or,
alternatively, in about 50 years, a “one-minute leap.”
Much more recently, a co-organizer of the 2011 Discussion Meeting “UTC for the 21st Centu-
ry” tendered the leap minute as a promising compromise.6 At this discussion meeting, attendees
were politely challenged by the Right Honourable David Willetts, Member of Parliament and the
Minister for Universities and Science, to discover an alternative to the current definition of UTC
that could still preserve a link between civil timekeeping and Earth rotation. Although the leap
minute was mentioned during discussions, it failed to gain technical acceptance and thus no rec-
ord of its mention survived the organizers’ account of the meeting.7, 8 An attendee from National
Institute of Standards and Technology had already advised that “a minute is an intolerably long
period of time. The only advantage is that it pushes the problem so far into the future that no one
is worried about it.”9 Another attendee from the UK National Physical Laboratory acknowledged
“A century down the line, we’ll need to introduce a ‘leap minute’, and nobody has any sensible
arguments for why that won’t be a worse issue” compared to leap seconds.10
An Explicit Leap-Minute Proposal
A detailed leap-minute proposal was submitted in response to the 2011 IERS questionnaire by
Pere Planesas of the Observatorio Astronómico Nacional, in Madrid, Spain.11 The proposal in-
cluded these specific elements:
• The application of a leap minute should target a year when predicted (UTC−UT1) equals
60 seconds.
• The time of insertion should prefer June 30th, because this date less is disruptive than New
Year's Eve.
• The announcement of a leap minute should be made “several years” ahead, “strictly”
keeping observed (UTC−UT1) between 55.0 s and 65.5 s on the date of application.
No disadvantages from leap minutes were acknowledged in the proposal, but some advantages of
a leap minute were noted:
• Leap minutes keep UTC “close to” mean solar time, maintaining UTC’s name and status.
3
• ∆UT1 corrections would be used more to recover UT1, giving more visibility to those
who determine these corrections.
• The first leap minute would not occur for several decades, allowing all hardware, soft-
ware, and time-dissemination standards to adapt to the change of convention.
• Fewer adjustments would be required per century, allowing the difference between Inter-
national Atomic Time (TAI) and UTC to remain constant for decades.
• Leap minutes cope well with quadratic UT1 separation in the very long-term, and also
avoid the possibility of negative corrections.
Although Planesas’ leap-minute proposal is detailed, the requirements addressed by its speci-
fications are unclear. The introduction of a leap minute when (UTC−UT1) is ~60 s maximizes the
magnitude of (UTC−UT1) and pushes the intercalary minute as far as possible into the future.
The prescription that a leap minute must occur within a window of ±5 s is not seemingly based on
specific technical criteria, yet it limits the amount of advanced notice from decades to years. And
there is also no technical rationale for the notion that “UT1 plus one minute” is a viable substitute
for specifications calling for mean solar time at Greenwich. The proposal presumes that an inter-
calary minute could only occur at either the end of June 30th or December 31st UTC, though leap
seconds can be introduced at the end of any Gregorian calendar month.12 And, while leap minutes
might cope with quadratic UT1 separation better than, say, leap seconds, that technical aspect
may not be relevant for perhaps a millennium.
THE TIMING OF A LEAP MINUTE
The question of when to introduce a leap minute is mostly an inquiry about the future separa-
tion of Universal Time (UT1) from Terrestrial Time (TT)*, also known as ∆T:
∆T = TT − UT1 (1)
Forecasts of ∆T are sometimes used for planning the potential magnitude and frequency of inter-
calary clock adjustments, but such forecasts are notoriously inaccurate. For example, Essen
(1967) suggested that such differences should vary less than 20 seconds over the next two centu-
ries: 13
“Consider now the consequences of using the atomic time scale. The atomic unit has
been made as nearly as possible equal to the second of Ephemeris Time (ET). During the
200 years or so during which ET has been compared with UT, the maximum deviation of
the two scales amounted to ±20s. If the earth continues to behave in a similar way, and
the atomic scale is set to give the time of day as given by astronomical measurements,
then the time it records should not differ from UT by more than ±20s during the next 200
years.”
In hindsight, Essen’s prediction greatly underestimated the rate of separation, for it took less than
one-quarter century, rather than two centuries, for the difference between UT and atomic time to
deviate 20 seconds. More recently, Vincent Meens, chairman of the ITU-R Study Group respon-
sible for recommending a redefinition of UTC, reportedly said “The [rate of] deviation of the leap
*
The predecessor of TT was Ephemeris Time (ET).
4
second is about 1 minute and 30 seconds per century. The deviation for a millennium will be on
the order of 15 minutes.”*
The predictive statements of Essen and Meens seem to presume that ∆T increases linearly. In
contrast, and within the context of redefining UTC, Nelson et al. (2001)4 illustrated a parabolic
separation, consistent with a change in the excess length of day of 2.3 ms/cy (or ∆T change of
42 s/cy2). Such rapid separation is compatible with a theory of lunar tides by which tidal braking
results in the observed tidal acceleration of the Moon, without other torques or modeled changes
to the principal moment of the inertia of the Earth.14 However, this degree of change is not ob-
served in ∆T since the invention of the telescope, and significantly overstates the observed retar-
dation of UT1 in recent years. The early work of Clemence (1948) estimated a value much less:
the average deceleration in Earth rotation “over the past 2000 years is 29 seconds per century per
century” (a change in the excess length of day of 1.6 ms/cy), predicting that “[b]y A.D. 3000 the
accumulated effect will amount to something like half an hour.”15
Parabolic Approximation of ∆T
Parabolic representation of ∆T, often used for general trend analyses and extrapolation, as-
sumes constant deceleration of Earth rotation, or, deceleration that varies somewhat randomly
around a single mean value. This is equivalent to saying that the rate of change in the excess
length of day (LoD), a representation of the time derivative of ∆T, is asymptotically constant.
However, parabolic fits highly depend on the data or theory adopted. For a parabola mathemati-
cally expressed as:
∆Tmodeled = a + c (t − t0)2 , (2)
the most influential information regarding the shape parameter c is available when (t − t0) is large.
However, over the telescopic era for which ∆T has been observed precisely, (t − t0) is compara-
tively small.
Phenomena not modeled by a constant rotational deceleration cause inaccurate extrapolations.
Long-term changes to the total angular momentum or mass moment of inertia of the figure of the
Earth, and/or changes in the differential rates between internal parts of the Earth (e.g., core v.
mantle), complicate the supposed parabolic shape of ∆T. The local change-point in the behavior
of ∆T at the end of the 20th century, and its subsequent dearth of leap seconds, is a recent remind-
er that the rotation rate of Earth has an unpredictably random component. For ancient observa-
tions before the invention of the telescope, decadal variations in the terrestrial rate of rotation in-
troduce irregular fluctuations about the mean trend of ∆T, leading to inherent geophysical uncer-
tainty in ancient observations of at least ±20 seconds.16 Some authors have used mathematical
analysis of the historic ∆T series to suggest the possible existence of harmonics signatures in the
excess length of day, which has led to some speculation about the possibility of a future negative
leap second.17 Stephenson & Morrison (1995), while hypothesizing the existence of very long-
period harmonics, also recognize significant differences in near-term and long-term estimates of
length-of-day.18 Within statistical accuracy, Huber (2006) modeled the stochastic behavior of
LoD process with three components (a global Brownian motion process, decadal fluctuation, and
a 50-day oscillation), and hypothesized that the component of acceleration often attributed to
post-glacial rebound might also be explainable, in principle, as a purely random affect.19
*
http://www.abc.net.au/science/articles/2012/01/18/3410293.htm
5Thus, parabolic approximation of ∆T is a relatively naïve approach to predicting the rotation
of Earth, even though this model is often employed for long-term extrapolation. Various parabol-
ic estimates illustrate the significant uncertainty of a parabolic approach by producing a wide
range of future values that could predict the insertion of a leap minute into UTC (Figure 1).
Figure 1. Observed ∆T With Parabolic Approximations.
Very Long-Term Approximations of ∆T
Morrison & Stephenson (2004).20 Stephenson, Morrison, and their colleagues have proposed
various long-term parabolic approximations for estimated ∆T series spanning several thousand
years, their work leveraging chronicled observations of ancient eclipses. For the tidal deceleration
of the Moon, their research culminated into the general recommendation that researchers use:
∆TM&S(2004) = −20 + 32 [(t − 1820.0)/100]2 s, (3)
for extrapolating beyond the limits of ∆T tables, assuming a lunar mean-motion rate of
n& = 26"/cy2.
McCarthy (2012).21 McCarthy explored the introduction of future intercalary seconds on a di-
vergent* yet predictable schedule. Consideration of a graphical illustration of the parabolic curve
attributed to “Mathews & Lambert” suggests that the model closely approximates:
*
“Divergent” means the proposed schedule does not constrain |UT1−UTC| to any specific value, as is the current prac-
tice, but tracks the extrapolation of a “parabolic fit based on geophysics” over the next five centuries.
6∆TMcC(2012) = −19 + 33.8 [(t − 1835.0)/100]2 s, (4)
The value of 33.8 s/cy2 in Eq. (4) is adapted from an estimate of 1.85 ms/cy from Mathews and
Lambert (2009) for the total rate of change in length of day.22 Mathews and Lambert assign a
standard error of ±1 ms/cy to the rate estimate.
Approximations of ∆T over the Telescopic Era
Espenak & Meeus (2006).23 Espenak & Meeus (2006) discuss the uncertainty in Earth’s rota-
tional period and its impact on the geographic visibility of eclipses in the past and future, using
higher-order polynomials over specific time intervals to approximate the behavior of ∆T. Their
extrapolation of ∆T from 2005 to 2050 is:
∆TE&M(2006) = 62.92 + 0.32217(t − 2000.0) + 0.005589(t − 2000.0)2 s, (5)
This expression is derived from a fit to predictions of ∆T in 2010 and 2050. The estimate at 2010
(66.9 s) was based on a linear extrapolation from 2005 using 39 s/cy (the average rate from 1995
to 2005). The estimate at 2050 (93 s) was linearly extrapolated from 2010 using 66 s/cy (the aver-
age rate from 1901 to 2000).
McCarthy & Babcock (1986).24 McCarthy & Babcock provide a useful reduction of ∆T and
length-of-day estimates based on observational data after 1656, including error estimates, assum-
ing a lunar deceleration rate of n& = 26"/cy. The following parabolic fit was determined:
∆TMcC&B(1985) = −5.156 + 13.3066 [(t − 1819.0)/100]2 s, (6)
with an uncertainty of ±0.3264 s/cy2 assigned to the quadratic term. An analysis by this author
determined that Eq. (6) is an unweighted least-squares curve fit to the semi-annual ∆T series from
1657.0 to 1984.5.
Weighted Least-Squares Parabola. The near-term predictive performance of Eq. (6) can be
made more accurate by weighting ∆T according to the estimated error of its series. A weighted
least-squares estimate using the same ∆T series and errors estimates from McCarthy & Babcock
(1986) over 1657.0-1984.5 yields:
∆TWeighted (1657-1985) = −1.73 + 20.8 [(t − 1826.5)/100]2 s . (7)
The value of the quadratic term of Eq. (7) falls well outside the uncertainty assigned by McCarthy
& Babcock to the same term from Eq. (6), but remains within the uncertainty assigned by
Mathews and Lambert (2009) to the quadratic term of Eq. (4).
If the observational series of McCarthy & Babcock (1986) is extended forward up to 2013.0,
and backward to 1630.0, a weighted solution becomes:
∆TWeighted (1630-2013) = −1.92 + 21.0 [(t − 1825.8)/100]2 s , (8)
For Eq. (8), semi-annual ∆T values from 1985.0 to 2013.0 were taken from USNO Series 7 tables
and assigned an uncertainty of 1 ms. From 1630-1655, ∆T values at five-year increments were
used from Stephenson & Morrison (1984)* and assigned an uncertainty of 20 s per Morrison &
Stephenson (2004).20, 25 The inclusion of six additional weighted data points from 1630-1655
*
This series is also available from Section K of the Astronomical Almanac.
7changed curve by ~0.3 s at 1650, and made a negligible change near the year 2000. Over the fit-
interval of several centuries, graphs of Eq. (7) and Eq. (8) appear practically identical. For the
purposes of this work, Eq. (8) is adopted and represented in all figures, this estimate having con-
sidered the most observational data.
Figure 2. ∆T during the 20th Century. A linear fit to ∆T over 1907-2013 is included,
from which TT-UT1 changed by one minute.
Degenerate Parabolæ. The historical sequence of ∆T back to 1907—over which ∆T changed
by one minute—does not generally follow the long-term parabolic trends ascribed by researchers
to lengthier historical series. Indeed, a linear fit to ∆T over the 20th century might seem just as
reasonable as any other low-order empirical approximation relative to the data available. The
coarse representation of ∆T as a sequence of degenerate parabolæ, or piecewise linear approxima-
tions, each spanning decades to centuries, seems viable in light of compelling work by Stephen-
son et al. (1997), which suggests, for example, a nearly linear trend in ∆T from mid-16th century
to the late 17th century. This trend accommodates a historically important record of a rare type of
solar eclipse* observed by Christoph Clavius in 1567, during a time where ∆T is otherwise poorly
established before the availability of telescopic observations.26 Other piecewise linear approxima-
tions might be imagined also spanning the 18th and 19th centuries, and again spanning the 20th
century. A linear fit implies that Earth’s mean rotation rate stays approximately constant over the
time interval to which it is fit. The advantage of piecewise approximation is that it does not at-
tempt to assign one deceleration value throughout history like a single parabola.
*
The apparent diameters of the Moon and Sun were so well matched that the eclipse was neither total nor annular,
rather, sunlight remained passing through lunar valleys.
8An unweighted linear trend, fit to semi-annual ∆T values from 1907.0 to 2013.0, results in:
∆TLinear (1907-2013) = −11.0 + 55.2 [(t − 2000.0)/100] s . (9)
The disadvantage of a naïve linear fit is that its predictability is questionable, because Earth’s ro-
tation can dramatically and permanently depart from the local tendency at any time. Nevertheless,
it seems useful to consider such an estimate to provide yet another trend for establishing a wider
perspective on the problem of ∆T predictability.
Leap-Minute Prediction by Extrapolation
The redefinition of UTC is a proposed agenda item for the ITU-R World Radio Conference
(WRC) of 2015. Draft recommendations have already proposed that UTC redefinition occur five
(5) years after adoption by the WRC; thus, A.D. 2020 is used here as a starting point for analysis.
The USNO currently estimates that ∆T will have reached 70 s at this starting date;* such that if
leap minutes become operational, then UTC with leap minutes (UTCLM) is presumed to revert to
an atomic scale equal to approximately (TT − 70) s at the time of adoption.
If ∆T tracks one of the trends in Figure 1, then Table 1 suggests the times at which each UT1
trend will have separated from (TT − 70) s. The estimated times are provided for separation val-
ues of 30 s, 60 s, and 90 s after 2020. (That is to say, these are dates at which extrapolations of ∆T
curves in Figure 1 and Figure 2 reach 100 s, 130 s, and 160 s.) The dates for ∆ = 30s suggest a
very wide range of potential dates for leap-minute insertion.
Table 1. Dates at which (TT since 2020 − 70 s) separate
from various ∆T trend approximations.
Year Year Year
Extrapolation ∆ = 30s ∆ = 60s ∆ = 90s
Morrison & Stephenson (2004) (2010) 2036 2057
McCarthy (2012) 2022 2045 2064
Weighted Parabola Fit (1630-2013) 2046 2076 2103
Espenak & Meeus (2006) 2057 2084 2106
Linear Fit (1907-2013) 2067 2120 2173
McCarthy & Babcock (1986) 2086 2125 2160
Another predictive approach might assume that UT1 will simply increase according to the
long-term trend at the time of changeover from leap seconds to leap minutes. See, for example,
the work of Steve Allen†, and McCarthy & Klepczynski (1999).27 This approach is illustrated in
Figure 3 by simply shifting various parabolic trends vertically to align them with the expected
value of ∆T at 2020, and then extrapolating the parabolic trend. The advantage of this approach is
that it forces all predictions to begin from the same origin of adoption for comparative purposes.
However, if a particular representation of the long-term trend of ∆T is asymptotically correct,
then the shifted prediction will be biased relative to the asymptotic behavior.
*
ftp://maia.usno.navy.mil/ser7/deltat.preds
†
http://www.ucolick.org/~sla/leapsecs/dutc.html
9Figure 3. Forecasts Based on Mean-Trend Accelerations. Post-2013 UT based on USNO forecast.
The dating results from Figure 3 are provided in Table 2. Added to Figure 3 are long-term
standard-error bounds due to Huber (2006), which are primarily based on an assumed global
Brownian motion process. These standard-error bounds are drawn relative to the weighted least-
squares parabola. (This curve was chosen because it is generally central to all the other curves.)
Although the Huber standard-error bounds cannot be considered an exact guide, they serve to il-
lustrate that such extrapolations, which graphically appear very different, may be statistically no
different from each other to within reasonable confidence levels. That is to say, various differ-
ences in long-term prediction may have less to do with changes in the long-term rate of accelera-
tion, and more to do with estimation uncertainty caused by random fluctuations in fitted ∆T. Ac-
cording to Table 2 then, one may conclude that a wide range of possible dates could be legiti-
mately assigned to first leap minute.
Table 2. Dates since 2020 at which UT1 separates from (TT−70) s
from various accelerating UT1 Forecasts.
Excess LoD Year Year Year
Extrapolated Curve Growth (ms/cy) ∆ = 30s ∆ = 60s ∆ = 90s
Morrison & Stephenson (2004) 1.75 2042 2062 2081
McCarthy (2012) 1.85 2042 2063 2081
Weighted Parabolic Fit (1630-2013) 1.15 2053 2083 2109
Espenak & Meeus (2006) 3.06 2057 2084 2106
McCarthy & Babcock (1986) 0.72 2069 2111 2147
Linear Fit (1907-2013) 0.00 2072 2125 (2178)
10Designating the Time of Century
Because of the wide range of possible dates at which a leap minute might occur, it becomes
practically impossible to assign far into the future the optimal date which would guarantee, say,
|UT1−UTCLM| to be “thirty-something” seconds. Thus, the adoption of a small threshold for
|UT1−UTCLM| would preclude assignment of a specific date for the first leap minute at the time
the new scheme is adopted. Such a scenario negates much of the presumed benefit of the proposal
and raises questions as to the scheme’s future success, because a supposed criterion to establish
the date of the first adjustment could be changed (or abolished) after the fact. The recourse would
adopt an insertion date based on some presumed behavior for UT1 far in advance. Minimizing
|UT1−UTCLM| by introducing a leap minute at ~30 s suggests an insertion date generally near the
middle of the 21st century, according to Table 2.
Designating the Time of Year
Another factor that comes into consideration is the time of year at which an adjustment should
occur. Electronic traffic and commerce will almost certainly be disrupted by the fact that a leap
minute would need to maintain an atypical representation such as 23:60:--, just as leap seconds
are labeled 23:59:60. A date should therefore be chosen to minimize its impact on commerce.
This suggests that a leap minute should occur on the equivalent of an international holiday.*
The Gregorian system, or Western calendar, is used extensively today a de facto international
standard.28, 29 Consequently, the first day of the Gregorian calendar, January 1, is now observed
almost universally as a public holiday.30 Experience suggests that some technicians find various
time-representation glitches caused by end-of-year roll-over, so increased awareness of potential
problems may already exist at this time of year more than any other. Also, recent history has
shown that many people expect leap seconds to occur at the end of the year, perhaps because the
end-of-year adjustments receive more media attention around the New Year’s Eve holiday. There
might be a strong sociological argument for maintaining intercalary adjustments at the end of the
year, considering that New Year’s Eve is mainly a celebration of the passage of time.
The disadvantage or inconvenience of choosing a holiday is that it requires more experienced
personnel monitoring potential technical issues that may arise from such a rare event, during a
time that traditionally would be spent away from regular duties. Also, more and more commerce
is being practiced on holidays, especially electronically, and thus the technical advantages of a
holiday date may wane with time. An alternative choice might associate the leap minute with
February 29th, which emphasizes that the leap minute is a long-term correction to the astronomi-
cal calendar.
Designating the Time of Week
Another consideration is the day of week on which an adjustment might fall, as there may be
some debate as to whether a weekday or a weekend would be preferable. Presuming that western
industrial conventions for a Monday-through-Friday work week are still being practiced decades
from now, the weekend may be a time when commerce and data collection would be least dis-
rupted by a calendrical adjustment. The final moments of Saturday on the meridian of Greenwich
would result in the leap minute occurring on Sunday for eastern half of the globe, and Saturday
for the western half. However, many Middle-Eastern countries currently practice a Fri-
*
This is in contrast with Planesas’ suggestion that the leap-minute insertion date avoid New Year’s Eve.
11day/Saturday rest period. Also, it is not clear if some cultures would object to the perception of
having to supply any extra manpower during a time period that is normally dedicated to rest.
Table 3. Day of Week for February 29th, 2020-2092
Sunday Monday Tuesday Wednesday Thursday Friday Saturday
2032 2044 2028 2040 2024 2036 2020
2060 2072 2056 2068 2052 2064 2048
2088 2084 2096 2080 2092 2076
Table 4. Day of Week for December 31st, 2018-2089
Sunday Monday Tuesday Wednesday Thursday Friday Saturday
2018 2019 2020 2021 2022
2023 2024 2025 2026 2027
2028 2029 2030 2031 2032 2033
2034 2035 2036 2037 2038 2039
2040 2041 2042 2043 2044
2045 2046 2047 2048 2049 2050
2051 2052 2053 2054 2055
2056 2057 2058 2059 2060 2061
2062 2063 2064 2065 2066 2067
2068 2069 2070 2071 2072
2073 2074 2075 2076 2077 2078
2079 2080 2081 2082 2083
2084 2085 2086 2087 2088 2089
Memorability of the Date
Table 3 and Table 4 tabulate the days of the week during much of the 21st century on which
the dates of February 29th and December 31st fall. If primary consideration is given to an insertion
date near the middle of the 21st century, and if weekend insertion is also given priority, then the
dates of February 29, 2048 and December 31, 2050 appear promising according to a “Satur-
day/Sunday weekend” preference, or December 31, 2049/January 1, 2050 according to a “Fri-
day/Saturday weekend” preference. However, when scheduling a rare event far into the future,
the memorability of the date may also play an important role. More significance may be attached
to a mid-century date like December 31, 2049, compared to, say, February 29, 2048. Traditional-
ly, the fundamental time epoch for astronomical theories have often been fixed to half-century
increments, such as 1800, 1850, 1900, 1950, and 2000.
12THE REPRESENTATION OF A LEAP MINUTE
There are three basic means of reconciling atomic time and astronomical time of day. If the
constancy of unit duration is maintained, this results in intercalary adjustments having unconven-
tional representations (such as 23:60). Otherwise, units of duration must be varied or “stretched”
to fit traditional representations of time in software and hardware. Outside of stretching or leap-
ing, the only remaining option is to replace one atomic timescale with another once the difference
from astronomical time is out of tolerance, much like the Gregorian calendar replaced the Julian
calendar. Implementation of a leap minute could be approached with any one of these methods.
However, at this stage it is unclear how a proposed leap minute should work any differently than
the leap second works now. Specifically, lengthening a UTC “day” by inserting exactly sixty (60)
TAI seconds at the end of the UTC day seems to appeal as a default approach, rather than inser-
tion at any other hour of the day.
Eyeing the distant future, it seems unlikely that hardware manufacturers would build special-
ized circuitry, gearing, infrastructure, or provide other complex arrangements, to account for a
leap minute, and there is little expectation that people will discard and replace existing hardware
for a once-in-a-lifetime event. Simply taking devices temporarily out of service is likely to be an
attractive option, as there are likely to be many computer codes that would otherwise interpret
23:60:-- as 00:00:-- of the next day, etc. The duration of a leap minute is significant enough that
many relatively inaccurate timekeepers would not be able to ignore it; thus, awkward representa-
tional issues with leap minutes would be more widespread than with leap seconds.
Because there is an implicit requirement that the display of a leap minute must comply with
historic systems of time representation, it seems reasonable to exclude the possibility of represen-
tation as a sequence of conventional leap seconds labeled from 60 to 119, because any representa-
tion that requires three-digit representation would be incompatible with digital time-display for-
mats expecting no more than two-digits worth of seconds. Consequently, the interval of a leap
minute might be expected to have a digital representation of seconds lasting from zero through
59. Or, representational issues might be implemented as a temporary rate adjustment to local
hardware clocks (such as UTC rate slewing of the 1960’s, the recent Google “leap smear”,* or
UTC-SLS†).
THE REQUIREMENTS SURROUNDING LEAP MINUTES
The justifications often offered for a proposal are indicative of the inherent requirements to be
addressed. Recognizable advantages supposedly addressed by a leap minute have thus far includ-
ed: public concerns about the separation of clock time from the Sun, operational convenience, and
preservation of “Coordinated Universal Time” as a named timekeeping convention. Each of these
is now discussed in detail.
Public Concerns
Hudson (1967) noted that, for the leap minute specifically, “approximate epochal coherence
with the rising and setting of the sun would be retained, and there need be no fear of a radical de-
parture from solar time for ‘everyday’ purposes.”5 Specifically, there is natural variation in the
timing of celestial phenomena from year to year due the fact that the beginning of the calendar
year shifts with respect to the solar year by as much as ¾ of a day at addition of February 29th
*
http://googleblog.blogspot.com/2011/09/time-technology-and-leaping-seconds.html
†
http://www.cl.cam.ac.uk/~mgk25/time/utc-sls/
13(Figure 4). The physical radius of the Sun subtends about one-half degree of arc, such that solar
timing can also vary by about one minute of time depending on whether one is referencing the
limb of the Sun, or its center. The annual timings of sunrise and sunset are only published to a
precision of one minute in the Astronomical Almanac, and local atmospheric conditions, terrain
obstructions, and changes in observer elevation, can add significantly to the unpredictably of ap-
parent rise and set times Thus, leap minutes appear to maintain the association of annual lighting
conditions and clock time at a level of precision useful for some “everyday” applications, while
also providing a symbolic mechanism to address public concerns that clock time should correlate
with Earth rotation.
Figure 4. Time of Astronomical Sunrise (McCormick Observatory, Charlottesville, Alt. 264 m
M.S.L.). Sunrise is when the center of the solar disc is 50' below the local horizon plane.
However, leap minutes do not offer any advantages over leap seconds with regard to supposed
public concerns over harmonization of clock with sky. Already, frequent leap seconds noticeably
provide “approximate epochal coherence” to a higher level of accuracy, and for a wider range of
technical applications. And because the coupling of civil time and Earth rotation is relaxed with
leap minutes, the symbolism is not as strong and thus potentially less satisfying.
Advanced Notice v. Frequency of Adjustment
There is an operational need for intercalary adjustments to be predicted “sufficiently” far in
advance. The debatable point is what constitutes “sufficiently”. Leap seconds tend to be an-
nounced six months in advance, but they could be announced earlier.31 UTCLM allows for ex-
tremely long-term announcement only if a small tolerance for |UT1−UTCLM| is not expected.
Very long-term announcement is not documented to be an operational requirement; instead, “suf-
ficient” frequency of adjustment is required. For example, Kamp (2011) argues that intercalary
adjustments “once every couple of years is not nearly often enough” to ensure that systems han-
dle them correctly.32 Huston (2012) echoes a similar opinion as it relates to leap-second bugs: “It
is often the case in systems maintenance that the more a bug is exercised the more likely it is that
the bug will be isolated and corrected.”33 An isolated leap minute very far in the future therefore
seems most inconvenient to the immediate testing requirements of developers, which is in stark
contrast to one of the primary academic arguments offered for its support, i.e., “[t]he idea behind
14the move to suppress the insertion of leap seconds is that less inconvenience would be caused if,
instead, a leap minute or so were inserted perhaps once every century.”6
“Advanced notice” also implies adequate infrastructure for promoting intercalary adjustments.
Intercalary adjustments are best transmitted directly to precise timekeeping devices in the short-
term.* Interestingly, many devices used for ‘everyday’ timekeeping (computers, cellular tele-
phones, etc.) are inherently poor clocks; they display time accurately because their internal oscil-
lators are frequently synchronized to time signals over communication networks. Because accu-
rate timekeepers tend to be connected to a networked time source, there is already infrastructure
in place to propagate intercalary adjustments electronically and automatically, given the appropri-
ate telecommunication protocols. Thus, the distribution of accurate, semi-frequent intercalary ad-
justments such as leap seconds is a telecommunication concern foremost. Relatively frequent
“conventionally scheduled”, or divergent, leap seconds could be announced in advance via the
same telecommunication protocols as accurately directed leap seconds are now, but their broad-
cast would not be as critical a concern for telecommunication, because their advanced scheduling
makes other avenues of distribution possible.
In contrast, it is quite unclear how a leap minute—scheduled decades in advance—would be
successfully “broadcast” to a future generation, because the announcement of a leap minute does
not appear to be a telecommunication issue. Adjustment mechanisms prescribed 100 or even 400
years into the future can be respected, evidenced by the successful employment of Gregorian cal-
endar reforms. However, a calendar is not a clock, and the algorithmic insertion of leap days is
not easily compared to a leap minute. Leap days benefit from regular and frequent recurrence,
and from having the customary representation of February 29th. A leap minute is without prece-
dent because of its novelty and peculiar representation—perhaps the calendrical equivalent of
proclaiming that a December 32nd should happen in the distant future.
Nomenclature
Another supposed advantage of the leap minute is that it supports continuing use of the term
“Coordinated Universal Time” for a scale that otherwise practically diverges from Universal
Time. This perspective may be supported from the somewhat abstruse fact that ‘Coordinated
Universal Time’ and ‘UTC’ originally referred to a broadcast convention which employed small-
er steps and slight rate adjustments during the 1960’s, yet was replaced by a new convention in
1972 which eliminated rate adjustments in favor of larger steps. Arguably then, the leap minute
continues an “existing tendency” to further loosen the tie between precise timekeeping and solar-
based time in a scale called ‘UTC’.34 However, this perspective neglects the fact that UTC was
always purposed to provide Universal Time to a precision of 0.1 s via both definitions: before
1972, this was accomplished primarily by frequency steering, whereas after 1971 it was accom-
plished with leap seconds plus digital encoding of the DUT1 correction. Leap minutes fail to sup-
ply Universal Time to any technically useful level outside of, say, gnomonics, and thus alter the
character and technical purpose of the civil-broadcast scale. This could lead to confusion as the
term “UTC” is repurposed to designate quite different concepts for time.35
Other Explanations Not Directly Supporting Requirements
Some justifications for preferring leap minutes do not completely support that option favora-
bly or logically. Some critical discussion follows of arguments often advocating leap minutes.
*
If there is too-little advanced notice, then the announcement may not be distributed broadly enough; if there is too-
much advanced notice, operators or devices risk losing track of a scheduled adjustment.
15Unnoticeable differences in the timing of celestial phenomena. A recognized position of advo-
cacy is that deviations by up to a minute in |UT1−UTCLM| will not affect the general public’s per-
ception of celestial phenomena. However, clocks already do not attempt to indicate apparent solar
time, mainly because the “average person” has no strong need to precisely predict celestial phe-
nomena outside of, for example, participation in specific ritual activities. Nonetheless, the effects
of a leap minute will be noticeable to those for which the relationships between celestial phenom-
ena, Earth rotation, and precise clock readings are technically important. Rather than the timing
of celestial events such as sunrise or sunset, a more general consideration for the public might be
how accurately clock readings should indicate the beginning and ending of the mean-solar day,
which has legal significance in many jurisdictions.
Greater publicity and/or cause for festivity. A leap minute would be a once-in-a-lifetime ex-
cuse for celebration and for heightened public curiosity in timekeeping. However, leap minutes
actually provide 59 fewer opportunities to “celebrate” and “publicize” versus the current defini-
tion of UTC. And, although there is a saying that “any publicity is good publicity,” the publicity
surrounding a future leap minute might not be celebrated due to the technical inconveniences it
might cause. Many “average people” were greatly inconvenienced either directly or indirectly by
the so-called “Y2K”* or “Millennium Bug”, which required significant resources to investigate
and remedy. Arguably, leap seconds receive favorable publicity—mostly as a curiosity—because
they have remained largely inconsequential.
Fewer adjustments ease the accounting. Timekeeping history is full of peculiarities such as ir-
regular months, local calendars, and holidays. Continuing simplification has not proven to be a
fundamental requirement, evidenced by changing legislation about daylight-saving/summer time,
zone-time jurisdictions, etc. The very arguments for breaking the tie between UTC and UT1 are
the same arguments that could be used to abolish zone time and daylight-saving time / summer
time—simplifications which have no widespread public advocacy. If accounting ease was the
ultimate goal of the leap minute, then one should expect the first leap minute to never be an-
nounced, because no adjustment makes for the simplest of all accounting. (An account into which
no withdrawals or deposits are made is forgotten.) Additionally, leap seconds will still need to be
accommodated in applications dealing with past data, with leap minutes complicating the histori-
cal accounting. Easy accounting thereby argues against the leap minute.
Leap minutes harmless to the “average person”. Unfortunately, the “average person” of the
distant future is hard to define in the present. Such persons will likely depend on advancing tech-
nology that could be ill-equipped for leap-minute adjustments in precise civil timekeeping. It
seems reasonable to suppose that leap minutes would be much more disruptive to later genera-
tions unaccustomed to global intercalary clock adjustments than leap seconds are to the present
generation. It also seems reasonable to suppose that leap minutes will be neglected in many cir-
cumstances, even if announced decades in advance. (The “Y2K” or “Millennium Bug” existed
despite universal knowledge that the year 2000 would come.) Any further loosening of the degree
of coupling between UT1 and UTC fosters a situation whereby realignment of civil time with the
rotation of the Earth becomes highly impractical because it fails to motivate system designers to
accommodate long-term adjustments.
*
“Y2K” (Year 2000) refers to systemic issues caused by the habitual representation of two-digit years in data
timestamps during the 20th century.
16CONCLUSION
To a casual spectator, the leap minute might seem like an elegant and logical compromise: it is
sixty times less frequent than the present convention of leap seconds, it is sixty times smaller in
magnitude than the previously proposed leap hour, and it addresses the requirement that civil
timekeeping maintain coherence with (mean) solar time. However, documented perspectives in-
dicate that the leap-minute approach has been in contention since the 1960’s and lacks consensus.
It is unclear what technical applications would be satisfied by a deviation of up to one minute be-
tween UT1 and UTC versus some other value, and an optimal insertion point cannot be predicted
very far in advance to great accuracy. Thus, the function of the leap minute seems entirely aes-
thetic, with precision time broadcasts becoming effectively decoupled from Earth rotation.
Although proximity to UT1 cannot be relied upon with a leap minute, other factors could es-
tablish an insertion date far in advance. These factors might include the memorability of the inser-
tion date, and supposed times of the year and days of the week which might minimize disruptions.
However, any significant change from the status quo will introduce some degree of confusion and
cause some existing systems to malfunction. A leap minute must still label events in an atypical
way, as with leap seconds, except a leap minute will be harder to ignore. Considering that as yet
“there is no equipment in the world that could handle a leap minute,”36 there is a real risk that a
distant adjustment scheme would not be reliably implemented when the declared time comes be-
cause the technicalities are pushed so far into the future that they would not be pragmatically ad-
dressed.37 Thus, the infrequency of the leap minute fails to meet a technological requirement to
maintain critical awareness necessary to support intercalary adjustments. The current system of
timekeeping therefore seems to meet existing and future requirements in ways that a leap minute
cannot, such that technologists should not act too hastily to alter or discard it.
ACKNOWLEDGEMENTS
The author is grateful to Steve Allen of UCO/Lick Observatory and to Ken Seidelmann of the
University of Virginia for helpful discussions, particularly with regard to UT1 forecasting.
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